The above assumptions on the constraint set and the objective function are standard in the literature on stochastic optimization, and in particular in the analysis of conditional gradient algorithms and multi-level optimization; see, for example, [1]}, [2]} and  [3]}. We emphasize here that the above smoothness assumptions are made only on the functions \((f_i)_{1\le i \le T}\)  and not on the stochastic functions \((G_i)_{1\le i \le T}\)  (which would be a stronger assumption). Moreover, the Lipschitz continuity of \(f_i\) 's are implied by the Assumption REF  and assuming the functions are continuously differentiable. However, for sake of illustration, we state both assumptions separately. In addition to these assumptions, we also make unbiasedness and bounded-variance assumptions on the stochastic first-order oracle. Due to its technical nature, we state the precise details later in Section  (see Assumption REF ).
